Scaling limits and regularity results for a class of Ginzburg-Landau systems

A NNALES DE L ’I. H. P., SECTION C

R OBERT L. J ERRARD

H ^ALIL M ^ETE S ^ONER

Scaling limits and regularity results for a class of Ginzburg-Landau systems

Annales de l’I. H. P., section C, tome 16, n

^o

4 (1999), p. 423-466

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Scaling limits and regularity ^results

for

a

class of Ginzburg-Landau systems

Robert L. JERRARD *

Department of Mathematics University of Illinois 1409 West Green Street Urbana, IL 61801, ^USA

Halil Mete

SONER ~

Department of Mathematics Carnegie ^Mellon University Pittsburgh, ^PA 15213, USA Vol. 16, ^n° 4, 1999, p. 423-466 Analyse ^{non linéaire}

ABSTRACT. - We

study

^{a class of}

parabolic systems

which includes the

Ginzburg-Landau

^{heat flow}

equation,

for uE :

R d R 2, as

^{well as some natural}

quasilinear generalizations

^for

functions

taking

values in ^> 2.

We prove that for solutions of the

general system,

^the

limiting support

^as

E -~ 0 of the energy ^measure is a codimension k manifold which evolves via mean curvature.

We also establish some local

regularity ^results

which hold

uniformly

in E. In

particular,

^{we establish a}

small-energy regulity

theorem for the

general system,

^{and we} prove ^a

stronger regularity

result for the usual

Ginzburg-Landau equation

^on

^R2.

^©

Elsevier, Paris

* Partially supported by ^the Army ^{Research Office and the National Science Foundation} through the Center for Nonlinear Analysis ^and by ^{the NSF} grant DMS-9200801.

t Partially supported by ^the Army Research Office and the National Science Foundation

through the Center for Nonlinear Analysis ^and by ^{the NSF} grants DMS-9200801, DMS-9500940, and by ^{the ARO} grant DAAH04-95-1-0226.

Annales de l’lnstitut Henri Poincaré - Analyse ^non linéaire - 0294-1449 Vol. 16/99/04/© Elsevier, Paris

424 ^{R. L.} JERRARD AND ^H. M. SONER

RESUME - Nous etudions ^une classe de

systemes paraboliques ^qui

comprennent 1’ equation

de chaleur

Ginzburg-Landau,

pour uE : ^Rd R2,

ainsi que ^des

generalisations quasilineaires

^{pour des}

fonctions

prenant

leurs valeurs dans

>

^2.

Nous

prouvons

que, pour ^les solutions du

systeme general,

^le

^support

limite

(lorsque

^{E ~}

⁰⁾

^{de mesure}

d’énergie

^{est une variete de}

codimension k qui

evolue selon sa courbure moyenne.

Nous etablissons en addition

quelques

^{resultats de}

regularite ^locale, qui

sont valides uniformement ^{en E. @}

Elsevier,

^Paris

1.

INTRODUCTION

We

p resent

ⁱⁿ this paper ^a

collection

of results

concerning

^the

asymptotic regularity

^and

qualitative ^behavior

of solutions of ^the

Ginzburg-Landau

system,

We also

propose

^and

study

^{a class of}

equations

^{which we believe are}

^natural generalizations

^of

^(1.1).

^These

^systems

^{have the form}

Here

Of

special

interest is the ^case where _p ⁼

k;

^{this is a direct}

generalization

of

(1.1).

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

The

Ginzburg-Landau system

arises in a

variety

of contexts,

including

models of

superconductivity

^{and of}

^systems

^of

coupled

oscillators near a

bifurcation

point,

^{see for}

example

^Kuramoto

[27]. Recently

^{the associated}

minimization

problem

^{has been studied in}

great

^detail

by Betheul, Brezis,

and Helein

[4], [5],

with refinements

by

^Struwe

^[26],

among others.

Neu

[19],

^{Pismen and} Rubinstein

[20],

Rubinstein

[21],

^E

[9],

^{and others}

have

analysed (1.1)

^{and the} associated

Schroedinger-type equation using

matched

asymptotic expansions.

^A number of results on the behavior of

(1.1)

in two space dimensions were obtained

by

^Lin

[17], [18].

We view

(1.2)

^{as a natural}

generalization

^of

^(1.1)

^to

energies

^with

nonquadratic growth

^{in the}

gradient

^{term. Given a solution uE of}

^(1.2)

we define

We think of EE as a energy

density

^{for the}

generalized Ginzburg-Landau

system.

^This

interpretation

^{is motivated}

by

the fact that

is

formally

^a

Lyapunov

functional for

(1.2).

We remark that

(1.2)

^{is not}

an

equation

^for

gradient

flow for the functional IE.

However,

^{it retains} many of the estimates satisfied

by (1.1),

estimates which are crucial to any

analysis

^of

properties

^of solutions.

(These

^{estimates are}

chiefly presented

in Section

2).

Also,

^{in the same} way that

(1.1)

^{is a} kind of model

problem

^for

codimension 2

pattern formation,

^the

generalized system (1.2)

can serve as a model

problem

^{for the}

study

^of

higher

codimension

pattern

^formation.

This view is

supported by

the results we

present

in Section

3,

^{which are} discussed

immediately

^below.

Our results fall into two classes.

First,

^we characterize the

qualitative

behavior of solutions of

(1.2)

in the limit as E -~

0,

^{in the case where} d

> 1~

⁼ _p. ^More

precisely, given

^a

family

of solutions uE of

(1.2)

^with

appropriate

^initial

^data,

^{we define an} associated

family

^{of measures}

v;,

^and

we show that the

support

^{of these measures, in the}

limit,

^forms

exactly

^a

( d - k)-dimensional

submanifold which evolves via codimension k ^mean

curvature

flow,

^at least for short times.

This

result,

^which

occupies

^Section

^3,

confirms the formal

computations

of Rubinstein

[21],

Pismen and Rubinstein

[20],

^{and E}

[9]

for the usual

Ginzburg-Landau system (1.1)

ⁱⁿ three space

dimensions,

^{and also}

applies

Vol. 16, n° 4-1999.

426 R. L. JERRARD AND H. M. SONER

to more

general

situations. It is

closely

^{related to a number of recent results}

about the

asymptotic

behavior of solutions of scalar

Ginzburg-Landau equations

and related

equations.

^For

example,

^Chen

[7], Evans,

^{Soner and}

Souganidis [11],

^Ilmanen

[23],

^{and Soner}

[23]

have shown that solutions of the Allen-Cahn

equation

^{in a}

singular

limit exhibit a

sharp

^interface

which evolves via codimension 1 mean curvature flow. The latter three papers establish this result

globally

ⁱⁿ

time, using

various weak notions of evolution via mean curvature.

Analagous

results have been established for more

general

scalar reaction-diffusion

equations by Barles,

^{Soner and}

Souganidis [2]

and Jerrard

[12],

among others.

The

larger part

of this paper is devoted ^to

establishing

^some

regularity

theorems. We first prove ^a small energy

regularity

result. In Section 4

we prove that if certain

weighted integrals

of the energy

density

^{EE are}

sufficiently small,

then EE is in fact bounded in some smaller

region.

This result is valid

uniformly

^for

parameter

^{values E E}

(0.1].

^Our

proof

uses a

monotonicity

^{formula and a Bochner}

inequality, following

^ideas

of Struwe

[24],

and Chen and Struwe

[8].

Small energy

regularity

^and

a

covering argument imply partial regularity results,

^as in Chen and Struwe

[8].

In the

special

^case of the usual

Ginzburg-Landau equation

ⁱⁿ

^R2

^x

[0, T],

we establish much

stronger regularity

results. We prove that if

integrals

^of

the energy

density

^are bounded in some

region,

then in fact the energy is

pointwise

bounded in a smaller

region.

^This

result,

^{which is}

again

^uniform

in E, follows from the small energy

regularity

^{via a}

blowup argument (Section 6)

^{and a}

Liouville-type

^theorem

(Section 7).

^The

blowup argument

is similar to one found in Struwe

[25].

This latter

regularity

result is used in another paper

by

^the

authors, [14]

in which we

completely

characterize the

asymptotic

behavior of solutions of

(1.1)

^{in H x}

[o, T ~,

^{where H C}

^R2

^{and T > 0. This}

^result,

^{which is}

valid

only locally

ⁱⁿ

^time, provides rigorous proof

of formal results of Neu

[19],

^E

[9]

and others.

The paper ^starts with ^a collection of estimates in Section 2.

One issue we do not address is the

solvability

^of

(1.2).

It is well- known that

(1.1)

admits smooth

solutions;

this follows from the work of

Ladyzhenskaya, Solonnikov,

and Uraltseva

[16],

^{as is} verified in

Bauman, Chen, Phillips,

^and

Sternberg [3],

^for

example.

Results of this sort are not so obvious in the case of the

generalized system (1.2).

^{It is not difficult to}

construct some sort of weak solutions of

(1.2),

^for

example by discretizing

in

time, solving implicitly

^at each time

step,

^and

passing

^to limits. To establish

regularity, ^however,

^{seems to}

require

^a

priori

^estimates.

Annales de l ’lnstitut Henri Poincaré - Analyse ^{non linéaire}

Such estimates are not, in

general,

^{valid for}

quasilinear systems,

^but

they normally

^{hold for}

systems

for which there is some sort of energy

density

which is itself a subsolution of an

elliptic

^or

parabolic equation.

^This

is the case for

(1.2),

^as is shown in

Proposition

2.1. It is therefore not unreasonable to

expect

^{that the same} estimate holds for

(1.2),

and thus that smooth solutions exist. In this paper,

however,

^{we focus on other issues}

and

simpy

^{assume the existence} of smooth solutions.

We will

always

^assume that the initial data for

(1.2)

^satisfies

Multiplying (1.2) by uE

^and

defining wE : - ~ 2,

^{we discover}

The maximum

principle

^thus

suggests

that any reasonable solution should

satisfy

for all

(x, t)

^E

^Rd

^x

[0, oo). Similarly,

estimates in Section 2

imply

^{that a}

well-behaved solution should have the

property

^that

Both of these statements will

hold, roughly speaking,

^as

long

^{as there is}

no influx of energy

from

⁼ +00. It is not hard to see, for

example,

that a solution

produced by

^the

implicit

^time discretization described above will have these

properties.

^We therefore further assume that for initial data as

described,

^{our solutions}

satisfy

^both

(1.6)

^and

(1.7).

^{To establish}

these estimates a

priori

^would

require

^{a delicate}

analysis

^and

might

^{not be}

possible,

^{as is shown}

by

^the

example

^{of the heat}

equation.

NOTATION AND

PRELIMINARIES

We will use the

following

^notation

throughout

this paper.

Integers d

and k will

always

denote the dimensions of the domain and the range,

respectively,

^{of the}

mappings

^{we consider.}

eE(.)

^and

E~(.)

^will

always

^{be as} defined in

(1.2)

^and

(1.3),

^{where the}

power p in the latter definition is understood to be the same as that in the

Vol. 16, n° 4-1999.

428 R. L. JERRARD AND H. M. SONER

generalized system (1.2).

^{We will}

normally

^{write eE} instead of

eE (uE ),

^when

no confusion can

result,

and likewise EE .

We

employ

the summation convention

throughout.

^{Roman indices}

i, j, , ...

are

always

understood to run from 1 to

d,

^and

greek

^indices a,

~,

^...

run from 1 to k.

Exceptions

will be indicated

explicitly.

^{A scalar}

product

between matrices is denoted

by

^{A :}

B,

^{so that for}

example

I .-

We also use the notation

We will

normally

^{omit the}

superscript n

which indicates the dimension of the ambient space,

displaying

^it

only

when the dimension is not obvious from the context.

Observe that if uE solves

(1.2)

^{for a}

given

value of the

parameter

^E, then

t)

^.-

uE(ax,

^solves

(1.2)

^{with e .-}

Similarly,

^we

have

t)

^_

a2t). Rescaling

^{in this}

fashion,

^{we can}

convert statements about solutions of

(1.2)

^for

arbitrary

^{E into statements}

about solutions with E ⁼

1,

^for

example.

^{Whenever a statement of a theorem}

is invariant under this

rescaling,

^it

clearly

^{suffices to} prove it for ^a

single

value of the

parameter

^{E. We will invoke this sort of}

argument

^{from time to} time

by saying,

without further

explanation,

that it suffices

"by

^a

rescaling argument"

^{to consider a certain case.}

2. ESTIMATES

In this section we collect some estimates that we will use

throughout

this paper.

We assume that uE is a smooth solution of

(1.2)

^on

Rd

x

[0,oo)

^and

that

EE(~, 0)

^E

L1 (R d).

Following

^a

suggestion

of M. Grillakis we define

The

following

fundamental identities are immediate consequences of the

equation (1.2).

^{We have}

Annales de l’lnstitut Henri Poincaré - Analyse ^{non linéaire}

Given a smooth test function _{yy E} x

~0, oo~ ),

^we

multiply

^{the first}

equation

^above

by ~

and the second

by Vyy,

then subtract to obtain

We

integrate

^{to find}

By adding,

rather than

subtracting, equations (2.1)

^and

(2.2),

^{we obtain}

in a similar fashion

The

integration by parts

^{that we} have carried out above is

justified

^if

The former follows from our

standing assumption ^(1.7). Invoking

^{the same}

assumption,

the latter holds for

a.e. t,

^since

and the

right-hand

^{side is finite a.e. t. Whenever we}

apply

^{the above}

estimates,

^{we will}

integrate

^{them over some time}

interval,

^{so we can}

safely ignore

^{the set of} measure zero on which

pE ( ~, t)

^{is not}

integrable.

We next show that the energy

density

^{EE solves a certain}

parabolic equation.

^{In the statement and}

proof

^of this lemma we omit all

superscripts

^E,

and we write e to mean

e(u)

⁼

eE(uE).

PROPOSITION 2.1. - The _energy

density

^E

satisfies

Vol. 16, n ° 4-1999.

430 R. L. JERRARD AND H. M. SONER

Also,

Proof -

^{From the} definition of E we

compute

We now

replace

^{ut and}

~ut

in the above

equation by expressions

^{we obtain}

from the

generalized Ginzburg-Landau system (1.2), thereby obtaining

(We

have written out

explicitly

^{the terms} for which there is some chance that more condensed notation

might

^be

ambiguous.)

^{We also have}

from which we deduce that

From these we

obtain,

^after

cancelling

^{several terms and}

combining

^terms

of the same

form,

Annales de l ’lnstitut Henri Poincaré - Analyse ^{non linéaire}

From the

definition

^{of e we see that}

The above ^two

equations immediately imply

^that

(2.5)

^holds.

To

prove (2.6)

^from

(2.5),

^{note that}

Cauchy’s inequality ^gives

If

lul2

^>

1/2

then the first ^{term in} the

right-hand

^{side is}

^{negative. If,}

^on

the other ^{hand, lul2 1/2}

^then

^{(1 - ~u~2)2 4(1 - Therefore}

With (2.5) ^this immediately yields ^(2.6).

^D

Finally

^we

^derive

^{some L-}

^bounds

for the energy. As these

bounds

depend

^{on ., we}

^again

^indicate

explicitly

^the

parameter .

in what follows.

From

(2.5)

^we

easily

^{see that :=}

^satisfies

~-~

Thus the

maximum principle ^implies

^{that for any}

smooth solution

^{use and} for all

s,t

^>

^0,

If we

strengthen

^our

assumptions

^{on the}

initial data,

^we

obtain

the

following

more

useful

^result.

PROPOSITION 2.2. -

Let u~

6

C-(R’

^x

smooth solution

of (1.2 )

^{with p >}

^{2, such}

^that

The

conclusion

^{of the lemma follows}

easily

^from

^standard regularity ^theory

^{if p = 2.}

Proof -

^1.

By rescaling

^it

^suffices

^to

^consider

^{the case e}

= 1.

Vol. 16, n° 4-1999.

432 R. L. JERRARD AND H. M. SONER

Let w :=

lul2 ^{and 03C8}

^{:= E +}

K(w - 1),

^{where K > 03BA will be fixed}

below. For a smooth

function §

^let

Then

using (1.5)

^we

compute

^that

This with

(2.5) gives

where

2. We estimate

Hence

Annales de l ’lnstitut Henri Poincaré - Analyse ^{non linéaire}

on the set

{eP/2

^>

K(p - 2)}. Combining

^these

calculations,

^{we obtain}

Note that

if 03C8

^{> 2K then E >} 2K and thus

eP/2

^>

K(p - 2).

3. For _p >

2,

^{set ..}

There is a

K(p)

^{> 1 such that}

C(K, p)

^> 2K + 1 for all K >

K(p).

Moreover,

^{if K >}

K(p) and 03C8

^{2K +}

1,

^then

Therefore

by taking

^{K =}

K(p)

^{v x} in the definition of

~,

^we

get

4. If we

set § :_ ~

^V

2K,

^then

.C~

^{0 on}

~~

^{2K +}

1} ⁽ⁱⁿ

^{the sense}

of

viscosity solutions)

^and

~(x, 0) =

^{2K. Let}

and define

From

(2.7)

^we deduce that

c( . )

^is continuous and that

to

^{> 0.}

Also, ,C~

⁰

on

Rd

x

( 0, to )

^{and so} the maximum

principle implies

^{that if t}

to

^then

~(x, t) ~(x, 0) =

^{2K. Thus}

to

^{= +oo}

and §

^{2I~ on}

^R~

^x

[0, oo).

^D

3. CONVERGENCE TO CODIMENSION k MEAN CURVATURE FLOW

In this section we consider examine

asymptotic

behavior of solutions of the

generalized Ginzburg-Landau system

^{in the case d > 1~ =} p.

For this purpose, it is convenient ^to introduce the normalized measure

Vol. 16, n 4-1999.

434 R. L. JERRARD AND H. M. SONER

In the

following,

we assume that

ro

^{is a} smooth embedded

compact

(d - k)-dimensional

submanifold of

Rd,

^{and that} is a smooth codimension k mean curvature flow

starting

^from

fa,

^{for some T >} 0. We let

T ~ Rd

x

[0, T]

^{denote the set}

swept

^out

by f t,

^i.e.

Also,

^{we define}

Since r is smooth and

compact,

^{we can find a number} ~o > 0 and ^a smooth

function ~

^{such that}

and

Ambrosio and Soner

[ 1 establish

^several

properties

^{of the}

function 1 203B42

ⁱⁿ

a recent paper. Their results

immediately imply

^that ri has the

following properties:

THEOREM 3.1. - For

(x, t)

^{in a}

neighborhood of 0393,

the matrix

t)

has k

eigenvalues equal

^{to one, and each}

of

^the

remaining

^{d - k}

eigenvalues satisfies

^the

estimate t) ( Cb(x, t).

^In

particular, ~2r~(x, t)

^{is a}

projection

^{onto a} k-dimensional

subspace

^when

(x, t)

^{E T.}

^Moreover,

In

fact,

Ambrosio and Soner

[1] ] show

^{that for a smooth}

evolving

^manifold

as

above, t) gives

the normal

velocity

^{vector of}

rt

^at

(x, t)

^E

^r,

and

t) equals

^{the mean} curvature vector. so the above

equation precisely

characterizes smooth codimension k mean curvature flow.

We will ^use these results in the

following

^form:

COROLLARY 3.1. - For

any 03BE

^E

^Rd

^{and all}

(x, t) in a neighborhood of

^{r we have}

Annales de l’lnstitut Henri Poincaré - Analyse ^{non linéaire}

Also,

ri

satisfies

Proof. -

^The first assertion is immediate from the above characterization of the

eigenvalues

^{of near F.}

To

verify

the second

assertion,

^{first note that}

The first of these

equalities

^holds

because ~

^{attains its minimum on}

r,

^and

the second follows from the

description

^of in Theorem 3.1. If we let

~

⁼

A??,

^{we thus have}

(again using

^Theorem

3.1)

Given any

(x, t)

^E

^Rd

^x

[0, T],

^{we can}

find y

^E

rt

^{such that =}

8(x, t).

We then have

The

following

theorem is an easy consequence of these

properties

^of r~.

THEOREM 3.2. -

Suppose

^that

^{Rd -~ R~}

^{is a} smooth solution

of

the

generalized Ginzburg-Landau system (1.2)

^with p ⁼ k and initial data

hE(X) for

^which

as E --~ 0. Then

Proof. -

^{We use} the smooth

function ~

^defined above in the

weighted

energy estimate

(2.3). Dropping

^a

negative

^{term and}

using (3.2)

^{and the}

definition of we have

Vol. 16, n° ^4-1999.

436 R. L. JERRARD AND H. M. SONER

Select s E

(0, ao]

^{such that}

(3.1)

^{holds on the set}

r:

⁼

t) s~.

^This

number s _{may be} chosen

uniformly

^{for t E}

[0, T),

^{so we} may ^assume that

~~~z~~~~~~1

^{C on}

R‘~ ~ Tt

^{for some constant}

^C, uniformly

ⁱⁿ

[0, T] .

^Then

Moreover,

The three

preceding inequalities together yield

Gronwall’s

inequality

^now

immediately implies

^that

for all 0 t ^T.

Dividing by log §,

^we obtain the conclusion of the

theorem. D ^.

Remark. - This

proof

may be ^{seen as a}

Pohozaev-type ^estimate,

^{as used}

for

example

ⁱⁿ

Bauman, Chen, Phillips,

^and

Sternberg [3].

The

hypotheses

of the above theorem mention

only

the initial distribution of energy. If ^{we assume} in addition that hE exhibits a vortex-like structure

along

cross-sections of

Fo,

^{so that}

Fo

^{is a}

"topological defect",

^{then we}

can strenthen the above result.

Because

ht

is assumed to be a smooth codimension k

manifold,

^{at each}

y ^E

rt

^we _{may find} ^vectors

nl(y, t),..., nk(y, t)

^E

^Rd

such that each na is normal to

Tt,

^and

n(3

⁼

8a(3.

^We assume moreover that

r t

is

orientable,

^{so that}

(y, t)

^~--~

na (y, t)

may be taken ^to be smooth and

globally

well-defined on r.

For y

^E

rt,

^{we define}

wE ( ~; ~, t) : ^{R~ --~ R~} by

Annales de l’Institut Henri Poincaré - Analyse ^non linéaire

THEOREM 3.3. -

Suppose

^{that uE is a} smooth solution

of (1.2)

^{with p = k}

and initial data uE

(x, 0)

^{= hE}

(x) satisfying

uniformly for

^{E > 0.}

If

ⁱⁿ addition there exists _{x, ~1 >} 0 such that

K/e

^and

for all y

^{E E then}

for all ~

^E

Co (Rd).

The constant comes from Lemma 3.2 below.

Remark. - Theorems 3.2 and 3.3 taken

together imply

^{that if vt} is any weak limit of

v;,

^{then the}

support of vt exactly equals ^{rt .}

As the

sign

^of

deg(wE ( . ;

^y,

t) ) ^depends

^{of the choice of ...,}

n k,

^we

may take it ^to be

positive,

without any loss of

generality.

We assume a > 0 is fixed and we introduce

the

^notation

We denote

typical points

ⁱⁿ

B4u

^{and K as x}

and y respectively.

^{Given a}

function vE : U ~

R/B

^{we further define}

Here

leb1

denotes 1-dimensional

lebesgue

^measure. We may think of

YtE

as the subset of

points

^{in K at} which the cross-section at time t exhibits

an isolated vortex, in ^a weak ^sense.

The

following

^{two estimates are} proven in Jerrard

[13]

^as Theorem 6.1 and Theorem 4.2

respectively.

Vol. 16, n° ^4-1999.

438 R. L. JERRARD AND H. M. SONER

for

^{some x >}

^0,

^{and assume that}

for all y

^E

K,

^{r E}

[03C3, 403C3],

^{and that}

for

^{all t E}

~0, T].

^Then

for

^{all t E}

[0, T ~ .

LEMMA 3.2. -

Suppose

^that

~E

^{E and that}

and that

Then

The constant

K( k)

^is

given explicitly

^{in Jerrard}

[13].

Using

^{these we}

present

^the

Proof of

Theorem 3.3.

1. For cr E

(o, ~1]

^{to be}

determined,

^{define U as above.}

First we define a

map 03C8

^E

CX> (K

^x

[0, T]; r)

such that for every t E

[o, T], t)

^{is a}

diffeomorphism

^{of K onto a subset of}

Tt.

^{Now we}

define 03A8 : U x

[0, T]

^~

^{Rd by}

Note that for fixed

(y, t), W(.,

^y,

t)

maps

R~

onto the normal space ^to

Annales de l ’lnstitut Henri Poincaré - Analyse ^{non linéaire}

We may ^assume that

I

^C and that the

C-1

>

0,

^where

= det is the Jacobian of ~.

We also assume that a is small

enough

^{that ~ is} one-to-one.

Finally

^{we define}

vE(x, y, t)

^.-

2. We now

verify

that vE satisfies the

hypotheses

of Lemma 3.1 Since uE is assumed

smooth,

^{it is evident that the}

map t H vE { ~, ., t)

^{is continuous}

in the norm of

W 1 ~ °° .

We next

compute

Also,

it is clear that

so the condition on the

degree

^of

v~ ( ~, ~, 0)

^follows

immediately

^from

(3.5)

and our choice of cr.

Finally,

^{note that and so}

The final

equality

follows from

(3.4) by

^the calculation in the

proof

^of

Theorem 3.2.

3. Lemma 3.1 therefore asserts that

(3.8)

^holds.

We now

define, for y E rt

It is clear that we can find a finite collection of of the form described above such that

Vol. 16, n ° 4-1999.

440 R. L. JERRARD AND H. M. SONER

Thus

(3.8)

^and

(3.9) imply

^that

4. For x

sufficiently

^{close to}

Ft,

^let

p(x)

^E

^rt

^{be the}

unique point

^of

rt satisfying

Fix r

so

^{small that}

p(x)

is well-defined on

{~(-,~) 4o-}.

^{Note that for}

Y ⁶

~,

This is an immediate consequence of Lemma 3.2.

In the calculations

below, Jp

^denotes the Jacobian of _p,

Jp

^:=

[det

^Here

dp

^{denotes the}

gradient

^of p considered as a map from

Rd

into and thus is

expressed

^{as a}

(d - k)

^{x d}

matrix,

^after

choosing

^{bases for the}

respective tangent

spaces. In

particular,

with this definition the

change

^{of variables that we}

employ

^{below is valid.}

For every

smooth, compactly supported §

^{we have}

where

and, by

^a version of the co-area

formula,

In the last

step

^{we have used}

(3.11).

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

5. Fix any

subsequence

^and a measure v such that ^--~ v.

By

Theorem

3.2,

^{we know that}

spt v

^C

ft.

^It follows that

We will show in Lemma 3.3 below that

Jp(y) =

^{1 for y E}

^ft.

^Thus

^Ii

vanishes as E ^~ 0.

Also, (3.12)

^and

(3.10) evidently imply

^that

LEMMA 3.3. - For _{p as}

defined

^above

and y

^E

^Ft, Jp(y)

^{= 1.}

Proof. - Fix y ~ 0393t

and orthonormal vectors To , ... , Td-k which _span

Ty rt . Taking

the standard basis e 1, ... , ^{ed as a} basis for

Ty Rd,

^{the matrix}

dp

has the form

After a

relabelling

^we may ^assume that _{ei == Ti} for i ⁼

1,...,

^{d - k and}

that are normal to

ft

^at y.

We claim that

Indeed,

for any

i = 1, ... , d - ~, by

the definition of _p,

since V6 is normal to

rt.

^Since

p(y)

^{= y, this}

implies

^that

which

implies (3.13).

Also, for j

^> d - k and h

sufficiently small,

^similar

reasoning

^shows

that

p(y

⁺

hej)

⁼

p(y).

^Thus

(dp)ij

^{= 0}

whenever j

^{> d - k. With}

(3.13)

and the definition of

Jp,

^this

implies

the conclusion of the lemma. D

Vol. 16, n° 4-1999.

442 R. L. JERRARD AND H. M. SONER

In the remainder of this

section,

^we

briefly

^{indicate a} way to construct

initial data hE for

(1.2)

^{in such a} way that the

resulting solutions,

^if

smooth,

will

satisfy

^the

hypotheses

^{of Theorems} 3.2 and 3.3.

We

impose

^some

topological

restrictions on

ro by assuming

^{that there}

exist

smooth, bounded,

open ^{sets =}

l,

...,

k,

such that

For a ⁼

1,..., ~,

^let

da

be the

(signed)

^{distance to so that}

Since the sets

C~a

^{are assumed}

smooth,

^{each function}

da

is smooth

near We assume in addition to

(3.14)

^that

on ro.

For ex ⁼

1,..., k,

^{let da be smooth} functions such that

Let d :

R~

be the vector-valued function whose ath

component

^is da . Note that d is related to the

ordinary

^{distance function}

b( ~, 0),

^defined

above, by

Finally,

^{note that}

assumption (3.14) implies

^{that -}

k-1/2(1, ..., 1)

as

]

^{-~ oo, so we} may find d

satisfying

^{the above}

conditions,

^{for which}

there exists some number M such that

Remark. - 1.

Assumption (3.14)

appears ^to be ^a necessary condition for the existence of initial data with the

required properties.

^Given

(3.17),

^one

can

modify

^{the sets}

^Oa locally

^near

ro

^to arrange that

(3.15)

be satisfied.

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

2.

Assumption (3.14)

is satisfied

by

any

Fo

^{which can} be embedded as a codimension 1 manifold in

Also,

^{it is}

clearly preserved

^under

homotopy.

Let v :

Rk R~

be a function of the form

v(x)

^{= for a}

scalar function _p such that

Then for

e(v)

⁺

W(v),

^{we have}

We define

One can then

verify

^that

in the sense of

distributions,

^and it is clear that

(3.5)

^holds.

Moreover,

^one

can

verify

that hE satisfies the

hypotheses

^of

Proposition 2.2,

and thus that

a smooth solution uE with hE as initial data satisfies

So hE has all of the desired

properties.

4. SMALL ENERGY REGULARITY

In this section we establish a small energy

regularity

theorem for solutions of the

generalized Ginzburg-Landau system.

^{The basic}

argument

we follow was introduced

by

^Schoen

[22]

^for

stationary

harmonic maps and

generalized by

^Struwe

[24]

^{and Chen and Struwe}

[8]

^{to the case of}

heat flow for harmonic maps and for

Ginzburg-Landau type approximations

of harmonic _maps.

The

proof

^{relies on a}

monotonicity

^{lemma and a}

Bochner-type inequality,

that

is,

^a differential

inequality

which is satisfied

by

the energy. The main

Vol. 16, n° 4-1999.

444 R. L. JERRARD AND H. M. SONER

novelty

^{here is the} observation that these estimates are available in this

more

general

context, ^{as well as the} fact that our result is local in nature. In

problems involving asymptotic

behavior of solutions of

Ginzburg-Landau

type systems, global

energy

estimates, independent

^{of E,}

typically

^{do not}

hold. Thus the local character of our estimates _{is very} useful in these

applications.

Small _energy

regularity

results of the sort that we establish here can

be used with

covering arguments

^{to deduce}

partial regularity results,

^{as in}

Chen and Struwe

[8].

We start

by establishing

^a

monotonicity formula,

^{which we}

get by putting

an

appropriate

^test

function ~

^{in the}

identity (2.5).

^{We first} define this function:

Let f : [0, oo)

^-~

[0,1]

^{be a smooth}

nonincreasing

^{function such that}

Also, define p : ^Rd

^x

(0, oo)

^{- R}

by

where I is the

identity

matrix. We then have

Annales de l ’lnstitut Henri Poincaré - Analyse ^{non linéaire}

So with this choice of _7y in

(2.4)

^{we obtain}

(using

^{the fact}

that ~

^is

nonnegative)

By

^the definition of _~y, the

integrals

^{on the}

right-hand

side above are

supported

^on

{x : 1~4 ~x~ 1~2}. ^Recalling

^that

~~~y~2/y

we have

Also,

^for

Ixl

^>

1/4,

Thus we have established the

following

^local

monotonicity

^formula.

LEMMA 4.1. - The measures

satisfy

the estimate

Before

stating

^our

small-energy regularity result,

^{we introduce some}

notation. For Xo E

Rd,

^{r >}

0,

^{and 0 t}

to,

^let

where q

^and p ^are defined at the

beginning

of this section. We write at to mean

ai.

Note that the

ar

^is scale-invariant in the

following

Vol. 16, nO 4-1999.

446 R. L. JERRARD AND H. M. SONER

sense: Given a function uE

solving (1.2),

^we _may define a rescaled function

by ic(x, t)

⁼

_uE(xo

+

Rx, R2t).

^We also define E ⁼

E(x, t)

⁼

2/p(eE(u))P~2.

^{As remarked in the}

introduction,

^{ic solves} the

system (1.2)

^with

scaling E,

^and

Thus, using

the fact that

p(Ry, s)

^{= we obtain}

_by

^a

change

^{of variables}

In

particular, by taking R

^{= r we can convert} statements about

ar

^to

statements about aE .

We now

change notation, using ~

^{to denote a small constant} which will be chosen below. We also introduce the notation

We now have

THEOREM 4.1

(local small-energy regularity). - Suppose

^{that uE is a}

solution

of (1.2)

^on

Br

^x

[To,

^{with E r.}

Suppose

also that there exists ~ > 0 such

that for

^{all x E}

Br,

^{s r with} C

Br,

^{and all}

t E

[To,

^{we have}

Then there are

positive

^constants ri, po, and C such that

if

for

^some

(xo, to)

^E

Br /4

^x

[To

⁺

T2, Ti]

^{and T E}

(0,

^{then we have}

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

Proof. -

^{1. We first claim that it suffices to} establish the theorem under the

assumption

^{that T =}

r vfij. ^Indeed,

^if

^T2

^{then we define r}

by insisting

that T ⁼

f vfij.

^{Note that r r, and so}

7(~x~~r) y(~:c~)

^{for all x. Thus}

Clearly

^also

(4.2)

^{continues to hold if r is}

replaced by

r. We may then use r instead of ^r in the

proof,

and the desired

equality

^{will be satisfied.}

Next, by rescaling

^we may ^{set r =} 1. Thus we assume that

The

constant ~

^E

(0,1]

will be fixed at the end of the

proof.

^{After these}

normalizations, (4.2) implies

^that

2. For 03C10 ~

(0,1/4]

^{to be}

^chosen,

^let

Thus

E (x, t)

^{2P[ in Estimate}

^(2.6)

^now

implies

^that

Vol. 16, nO ^4-1999.

448 R. L. JERRARD AND H. M. SONER

Note also that it now suffices to show that C for

appropriate

choices of _{q, po.}

Indeed,

^{if we have} this

estimate,

^then

which is the conclusion of the theorem.

Then

and w in

P~ (x, I).

^The coefficients in the above

equation satisfy

so

by

^a

parabolic

^Harnack

inequality

^for

nondivergence

^structure

equations,

see

Krylov

^{and Safonov}

[15],

^which

depends only

^{on the} above bounds

on the

coefficients,

^{we have}

4.

Since ~7

^{1 and} po

1/4,

and so for

(x, t)

^E

~a (~, t),

^{we have}

Thus

by

^the

monotonicity formula,

^{Lemma 4.1.}

Annales de l’lnstitut Henri Poincaré - Analyse ^{non linéaire}

Recall that

by construction, t to,

^{so the last term on the}

right

^hand

side above is bounded

by

5.

By translation,

^we may ^set xo ⁼

0,

^{and we}

define i := t

+

~2 - _to.

Observe that

by

construction we have

We now claim that if _po is

sufficiently small,

^then

We write

where

Recalling that ==

^{1 on}

B1/4(0),

^{we have}

if

1/8

^>

H.

^which may be

achieved,

^for

given

q,

by adjusting

po.

Taking

po still smaller

yields Ii

To estimate note that if

p(x -

⁺

i) -

⁺

i)

^{> 0 then}

We rewrite this

inequality

^as

Vol. 16, n ° 4-1999.

450 R. L. JERRARD AND H. M. SONER

This

implies

^that

using (4.6).

^Thus

if _po is chosen small

enough.

^With

(4.3)

^this

implies

^that

I2 r~~3.

The estimate of

13

is very similar to that of

I2,

^{so we omit it. Thus}

we have _proven our claim.

6.

Putting together steps

^{4 and}

5,

^{we find that}

Taking ~

^{small now}

gives

As remarked in

step 2,

^this

immediately yields

^the conclusion of the theorem. D

5. SOME VARIATIONS

By modifying

^the

argument

of the small energy

regularity theorem,

^we

obtain

slightly

^{different results which will be} useful later on.

PROPOSITION 5.1. -

Suppose

that u~ solves

(l.l )

^on

Br

^{x where}

E r. Assume also that there exists ~ > 0 such that

Then there exist constants

C ( ~ ) , T()

^{such that}

Remark. - Note that this

applies only

^to the usual

Ginzburg-Landau

system

^with

quadratic growth.

Annales de l ’lnstitut Henri Poincaré - Analyse ^{non linéaire}

Proof. -

1. As in the

proof

^of

Proposition ^4.1,

^{it suffices to prove the}

result for r ⁼ 1 > E. We may also ^assume

by

^a translation that

To

^{= 0.}

Take Xo E

B1 /2, ^to 1 / 16

^{to be fixed}

^later,

^{and define}

Exactly

^{as in the}

proof

^of

Proposition

^{4.1 we select ao E}

(0, to)

^and

(x, t)

^E

Pao

^{such that}

We further define 3 :=

( to - Q~)/2. ^Following

^the

^argument

^of

^steps

^1-4

of the

proof

^of

Proposition 4.1,

^{we find that}

for some 3 3.

2.

By

^the

monotonicity lemma,

^the definition of aE and the assumed L°° bounds on

E~ ( ~, 0),

From the definitions we

have t + jj2 2to,

^{so with}

Step

^{1 we obtain}

So there exists some T > 0 such that if

to

T, then

Next define

Vol. 16, n° ^4-1999.